Дыхта В.А. Динамические системы в экономике. Введение в анализ одномерных моделей
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p x
0
p x
0
= f
p
(x
0
)
x
0
x = f
p
(x),
p f
p f
p
x
t+p
= x
t
t ≥ t
0
f(x) = x
2
x
0
= −1
{−1, 1, 1, 1, . . .},
t
0
= 1
{1}
¯x = 0 x
∗
=
λ − 1
λ
.
x ≥ 0 x
∗
> 0
λ > 1
= R
f
0
(¯x) = f
0
(0) = λ, f
0
(x
∗
) = 2 − λ,
¯x λ ∈ (0, 1)
λ > 1 x
∗
−1 < 2 − λ < 1
λ ∈ (1, 3) λ > 3 λ ∈ (0, 1)
0 < λ < 1
0
f(x)
x
x
∗
1
1
2
λ = 1
0
f(x)
x
1
1
2
1 < λ < 3
0
f(x)
x
x
∗
1
1
2
λ < 3
λ = 1 λ = 3
|f
0
(x)| < 1 |f
0
(x)| > 1
f(x)
x λ
f(x, λ) f
λ
(x)
λ > 0
f
λ
: →
λ
λ
x
t+1
= f
λ
(x
t
)
¯x x
∗
λ = 1 λ = 3
λ = 1 ¯x = 0
x
∗
¯x λ = 1
λ ≥ 1 x
0
> 1
x ∈ [0, 1]
λ = 3
x
∗
λ ∈ (1, 3)
λ > 3
λ
f
λ
λ
f
2
(x) = λ
2
x(1 − x)[1 − λx(1 − x)].
λ > 3 f
2
x
1
, x
2
∈ (0, 1)
{x
1
, x
2
, x
1
, x
2
, . . .} {x
2
, x
1
, x
2
, x
1
, . . .}
{x
1
, x
2
}
f(x)
f
2
(x)
1
f
2
(x) = x
λ
2
(1 − x)(1 − λx + λx
2
) − 1 = 0
x = x
∗
(x − x
∗
) =
µ
x −
λ − 1
λ
¶
=
λx − λ + 1
λ
,
(x − x
∗
)Q(x)
Q(x)
Q(x) = λ
2
x
2
− λ(λ + 1)x + λ + 1 = 0.
x
1,2
=
λ + 1
2λ
±
√
λ + 1
√
λ − 3
2λ
.
x
1
, x
2
∈ (0, x
∗
) λ > 3
f
f
2
f(x
1
) = x
2
f(x
2
) = x
1
.
f
2
(x
1
) = x
1
f
2
¡
f(x
1
)
¢
= f
¡
f
2
(x
1
)
¢
= f(x
1
)
f(x
1
) f
2
x
2
f(x
1
) 0
x
∗
f
2
(x
1
) = x
1
2 2
0
f(x)
x
x
0
˜x
0
1
0
f(x)
x
x
0
1
λ = 3, 2
λ = 3, 5
f
0
(x
∗
) < 0
{x
1
, x
2
}
f
2
x
t+2
= f
¡
f(x
t
)
¢
, (2.74)
f(x
t+1
)
x
1
x
2
{x
1
, x
2
}
λ = 3
x
∗
λ λ
(f
20
)(x
i
) =
d
dx
[f
2
(x
i
)], i = 1, 2
λ
2
= 3, 45
λ f
4
λ
x
1
, x
2
λ = 2, 8
a
2
λ = 3
a
2
λ = 3, 2
a
2
x
1
x
2
x
1
x
2
λ = 3
f
0
λ
x
∗
λ > λ
2
x
1
, x
2
f
4
λ = λ
2
λ
{λ
n
} λ λ = λ
n
2
n
2
n+1
λ
n
− λ
n−1
λ
n+1
− λ
n
≈ 4, 669201609 . . . .
{λ
n
}
1/δ δ = 4, 6692
f(x)
lim
n→∞
λ
n
− λ
n−1
λ
n+1
− λ
n
= δ, (2.75)
δ
x
0
= 0, 1 f
λ
f
n
λ
(x
0
)
λ λ
{λ
n
}
λ
∞
= lim
n→∞
λ
n
≈ 3, 569 .
λ = λ
∞
λ λ
∞
(λ
∞,
4]
λ ≈ 3, 83
p = 1, 2, 3, . . .
S
f : S → S f
p f q
q > p
, , , ,
2 · 3, 2 · 5, 2 · 7, 2 · 9, . . .
2
2
· 3, 2
2
· 5, 2
2
· 7, 2
2
· 9, . . .
2
3
· 3, 2
3
· 5, 2
3
· 7, 2
3
· 9, . . .
. . . , . . . , . . . , . . . , . . .
. . . , 2
p
, . . . , 2
2
, 2
1
, 1.
S
q = 1, 2, 3, . . . .
f
2
p
, . . . , 2
2
, 2
1
, 1
p
x
0
x
0
t
λ = 3, 7 > λ
∞
x
0
= 0, 20 x
0
= 0, 21
f
λ
• f x
f
p
(x) = x p > 0 f
p
p f
• p f
f
p
x
t+1
= f
p
(x
t
)
• p
x x
• p {x, f(x), f
2
(x), . . . ,
f
p−1
(x)} p x
•
•
•
•
•
x
t+1
= 2x
t
− 10x
2
t
x
t+1
=
1
2
x
2
t
− x
t
x
t+1
= ax
t
− bx
2
t
, a, b > 0.
b
f
p
4
f
λ
p =
1, 2, 3, . . . f
4
2
p
p
x
t+1
= 4x
t
(1 − x
t
).
f
λ
lim
p→∞
f
p
2
(x) =
1
2
, x ∈ (0, 1).